which proves the inductive step.

In order to study the regularity in the variable \(\theta \in \mathbb {S}^{d-1}\)  we follow the spirit of [1]} which consists in studying the regularity in the projected variable \(v\in \mathbb {R}^{d-1}\) . The main technical difficult lies in the treatments of weights that account for the geometry of the problem. Additionally, derivation in angle is an operation that does not commutes with the equation which complicates the computations, yet, easily solved with a commutator identity of the type given in Proposition REF .
